How can I make a view that seems like orthographic, but not exactly?
With some limitations (or I haven't found the how to, probably), we can tweak the camera matrix when using Gpu module. So that we can have the equivalent of viewport rendering.

The image above is taken with a orthographic camera placed in front of some cubes:

On bottom left the addon (see below) shows a vignette of the tweaked image.
When in camera view you can move the camera or the object and view the corresponding result of the projection:

To take a screenshot, use the D key.
Limitations:

The camera targets along Y and it Y position is fixed to -8 units
The rendering is limited to the 3D view size
Moving objects along Y may push them out of the view
Works in rendered viewport shading but for Eevee or Workbench only (not Cycles)

Principles of the script:

It is based on simple modal operator template
Uses this documentation example to make the rendering

The &quot;key&quot; part of the code: as we can give to the GPU module the matrices we want, have added empirical values to make the projection (offset and tweak below).
camera_matrix = scene.camera.matrix_world.copy()
# Static camera position in Y
camera_matrix[1][3] = -8

view_matrix = camera_matrix.inverted()

projection_matrix = scene.camera.calc_matrix_camera(
    context.evaluated_depsgraph_get(), x=width, y=height)

# Empirical offset and perspective tweaks
projection_matrix[0][3] = OFFSET_X
projection_matrix[1][3] = OFFSET_Y

projection_matrix[0][2] = TWEAK_X
projection_matrix[1][2] = TWEAK_Y

# Draw the 3D view    
offscreen.draw_view3d(
    scene,
    context.view_layer,
    context.space_data,
    context.region,
    view_matrix,
    projection_matrix)

Blend file: execute the script then search F3 for &quot;oblique&quot; in the 3D view. Esc to quit. D to take a screenshot. You can zoom in/out by scaling the camera.

The rendering itself is equivalent to &quot;viewport render image&quot; (menu &quot;view&quot;). So you can have the shading you want and disable gizmo and overlays if you want).